recipe for coconut milk yogurt without stabilizers

And about the cultures, I've used slightly different ones for cows milk yogurt but they mostly gave me similar results. The differences mainly came from using UHT vs pasteurized, milk powder vs none.
S thermophilus and L bulgaricus give a tangy taste I like though.

http://chocolateandzucchini.com/recipes/basics/homemade-coconut-milk-yogurt-recipe/

Here's a recipe that gives you the option of using gelatin or not. I haven't tried it though. She does say that the version without gelatin separates, but the texture is fine when she stirs it again.
From my experience of homemade cow's milk yogurt, I only get a thick and creamy texture if I use UHT (ultra heated) full-fat milk and add 1/4 cup of milk powder per 1 liter of milk. Maybe a similar process would help achieve that texture for coconut yogurt: you could add extra coconut cream ?
Also what is your source for coconut milk ? If it is the canned type, do know that most of them do contain stabilizers in the form of guar gum, usually, to prevent separation.